Let $g(x)=x^3+x^2+x$ Where does $g$ have critical points? Choose all answers that apply: Choose all answers that apply: (Choice A) A $x=-1$ (Choice B) B $x=-\dfrac{2}{3}$ (Choice C) C $x=0$ (Choice D) D $g$ has no critical points.
Solution: A critical point of $g$ is a point in the domain of $g$ where the derivative is either equal to zero or undefined. So in order to find the critical points of $g$, let's find its derivative. $\begin{aligned} g'(x)&=\dfrac{d}{dx}\left[ x^3+x^2+x \right] \\\\ &=3x^2+2x+1 \end{aligned}$ Now let's look for $x$ -values where $g'$ is zero or undefined. $3x^2+2x+1=0$ has no solution, so $g'$ is never equal to $0$. $3x^2+2x+1$ is never undefined, so $g'$ is never undefined. In conclusion, $g$ has no critical points.